We establish the existence of three new subgroups of the group of volume-preserving diffeomorphisms of a compact $n$-dimensional ($n\ge 2$) Riemannian manifold which are associated with the Dirichlet, Neumann, and Mixed type boundary conditions that arise in second-order elliptic PDEs. We prove that when endowed with the $H^s$ Hilbert-class topologies for $s>(n/2)+1$, these subgroups are $C^\infty$ differential manifolds. We consider these new diffeomorphism groups with an $H^1$-equivalent right invariant metric, and prove the existence of unique smooth geodesics $\eta(t,\cdot)$ of this metric, as well as existence and uniqueness of the Jacobi equations associated to this metric. Geodesics on these subgroups are, in fact, the flows of a time-dependent velocity vector field $u(t,x)$, so that $\partial_t\eta(t,\cdot) = u(t,\eta(t,\cdot))$ with $\eta(0,x)=x$, and remarkably the vector field $u(t,x)$ solves the so-called Lagrangian averaged Euler (LAE-$\alpha$) equations on $M$. These equations, and their viscous counterparts, the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations, model the motion of a fluid at scales larger than an a priori fixed parameter $\alpha >0$, while averaging (or filtering-out) the small scale motion, and this is achieved without the use of artificial viscosity. We prove that for divergence-free initial data satisfying $u=0$ on $\partial M$, the LAE-$\alpha$ equations are well-posed, globally when $n=2$. We also find the boundary conditions that make the LANS-$\alpha$ equations well-posed, globally when $n=3$, and prove that solutions of the LANS-$\alpha$ equations converge when $n=2,3$ for almost all $t$ in some fixed time interval $(0,T)$ in $H^s$, $s\in (n/2+1,3)$ to solutions of the LAE-$\alpha$ equations, thus confirming the scaling arguments of Barenblatt \&{} Chorin.